3.6.22 \(\int \frac {(a+c x^2)^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=123 \[ \frac {4 c (d+e x)^{3/2} \left (a e^2+3 c d^2\right )}{3 e^5}-\frac {8 c d \sqrt {d+e x} \left (a e^2+c d^2\right )}{e^5}-\frac {2 \left (a e^2+c d^2\right )^2}{e^5 \sqrt {d+e x}}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5}-\frac {8 c^2 d (d+e x)^{5/2}}{5 e^5} \]

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Rubi [A]  time = 0.05, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {697} \begin {gather*} \frac {4 c (d+e x)^{3/2} \left (a e^2+3 c d^2\right )}{3 e^5}-\frac {8 c d \sqrt {d+e x} \left (a e^2+c d^2\right )}{e^5}-\frac {2 \left (a e^2+c d^2\right )^2}{e^5 \sqrt {d+e x}}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5}-\frac {8 c^2 d (d+e x)^{5/2}}{5 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2/(d + e*x)^(3/2),x]

[Out]

(-2*(c*d^2 + a*e^2)^2)/(e^5*Sqrt[d + e*x]) - (8*c*d*(c*d^2 + a*e^2)*Sqrt[d + e*x])/e^5 + (4*c*(3*c*d^2 + a*e^2
)*(d + e*x)^(3/2))/(3*e^5) - (8*c^2*d*(d + e*x)^(5/2))/(5*e^5) + (2*c^2*(d + e*x)^(7/2))/(7*e^5)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^{3/2}}-\frac {4 c d \left (c d^2+a e^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 c \left (3 c d^2+a e^2\right ) \sqrt {d+e x}}{e^4}-\frac {4 c^2 d (d+e x)^{3/2}}{e^4}+\frac {c^2 (d+e x)^{5/2}}{e^4}\right ) \, dx\\ &=-\frac {2 \left (c d^2+a e^2\right )^2}{e^5 \sqrt {d+e x}}-\frac {8 c d \left (c d^2+a e^2\right ) \sqrt {d+e x}}{e^5}+\frac {4 c \left (3 c d^2+a e^2\right ) (d+e x)^{3/2}}{3 e^5}-\frac {8 c^2 d (d+e x)^{5/2}}{5 e^5}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 97, normalized size = 0.79 \begin {gather*} -\frac {2 \left (105 a^2 e^4+70 a c e^2 \left (8 d^2+4 d e x-e^2 x^2\right )+3 c^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )}{105 e^5 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^2/(d + e*x)^(3/2),x]

[Out]

(-2*(105*a^2*e^4 + 70*a*c*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) + 3*c^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d
*e^3*x^3 - 5*e^4*x^4)))/(105*e^5*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 0.06, size = 123, normalized size = 1.00 \begin {gather*} \frac {2 \left (-105 a^2 e^4-210 a c d^2 e^2-420 a c d e^2 (d+e x)+70 a c e^2 (d+e x)^2-105 c^2 d^4-420 c^2 d^3 (d+e x)+210 c^2 d^2 (d+e x)^2-84 c^2 d (d+e x)^3+15 c^2 (d+e x)^4\right )}{105 e^5 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + c*x^2)^2/(d + e*x)^(3/2),x]

[Out]

(2*(-105*c^2*d^4 - 210*a*c*d^2*e^2 - 105*a^2*e^4 - 420*c^2*d^3*(d + e*x) - 420*a*c*d*e^2*(d + e*x) + 210*c^2*d
^2*(d + e*x)^2 + 70*a*c*e^2*(d + e*x)^2 - 84*c^2*d*(d + e*x)^3 + 15*c^2*(d + e*x)^4))/(105*e^5*Sqrt[d + e*x])

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fricas [A]  time = 0.39, size = 117, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (15 \, c^{2} e^{4} x^{4} - 24 \, c^{2} d e^{3} x^{3} - 384 \, c^{2} d^{4} - 560 \, a c d^{2} e^{2} - 105 \, a^{2} e^{4} + 2 \, {\left (24 \, c^{2} d^{2} e^{2} + 35 \, a c e^{4}\right )} x^{2} - 8 \, {\left (24 \, c^{2} d^{3} e + 35 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{6} x + d e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

2/105*(15*c^2*e^4*x^4 - 24*c^2*d*e^3*x^3 - 384*c^2*d^4 - 560*a*c*d^2*e^2 - 105*a^2*e^4 + 2*(24*c^2*d^2*e^2 + 3
5*a*c*e^4)*x^2 - 8*(24*c^2*d^3*e + 35*a*c*d*e^3)*x)*sqrt(e*x + d)/(e^6*x + d*e^5)

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giac [A]  time = 0.20, size = 137, normalized size = 1.11 \begin {gather*} \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} e^{30} - 84 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d e^{30} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt {x e + d} c^{2} d^{3} e^{30} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a c e^{32} - 420 \, \sqrt {x e + d} a c d e^{32}\right )} e^{\left (-35\right )} - \frac {2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-5\right )}}{\sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/105*(15*(x*e + d)^(7/2)*c^2*e^30 - 84*(x*e + d)^(5/2)*c^2*d*e^30 + 210*(x*e + d)^(3/2)*c^2*d^2*e^30 - 420*sq
rt(x*e + d)*c^2*d^3*e^30 + 70*(x*e + d)^(3/2)*a*c*e^32 - 420*sqrt(x*e + d)*a*c*d*e^32)*e^(-35) - 2*(c^2*d^4 +
2*a*c*d^2*e^2 + a^2*e^4)*e^(-5)/sqrt(x*e + d)

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maple [A]  time = 0.05, size = 106, normalized size = 0.86 \begin {gather*} -\frac {2 \left (-15 c^{2} x^{4} e^{4}+24 c^{2} d \,x^{3} e^{3}-70 a c \,e^{4} x^{2}-48 c^{2} d^{2} e^{2} x^{2}+280 a c d \,e^{3} x +192 c^{2} d^{3} e x +105 a^{2} e^{4}+560 a c \,d^{2} e^{2}+384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^2/(e*x+d)^(3/2),x)

[Out]

-2/105/(e*x+d)^(1/2)*(-15*c^2*e^4*x^4+24*c^2*d*e^3*x^3-70*a*c*e^4*x^2-48*c^2*d^2*e^2*x^2+280*a*c*d*e^3*x+192*c
^2*d^3*e*x+105*a^2*e^4+560*a*c*d^2*e^2+384*c^2*d^4)/e^5

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maxima [A]  time = 1.35, size = 121, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (\frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} - 84 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d + 70 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 420 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} \sqrt {e x + d}}{e^{4}} - \frac {105 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}}{\sqrt {e x + d} e^{4}}\right )}}{105 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

2/105*((15*(e*x + d)^(7/2)*c^2 - 84*(e*x + d)^(5/2)*c^2*d + 70*(3*c^2*d^2 + a*c*e^2)*(e*x + d)^(3/2) - 420*(c^
2*d^3 + a*c*d*e^2)*sqrt(e*x + d))/e^4 - 105*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)/(sqrt(e*x + d)*e^4))/e

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mupad [B]  time = 0.05, size = 128, normalized size = 1.04 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}-\frac {\left (8\,c^2\,d^3+8\,a\,c\,d\,e^2\right )\,\sqrt {d+e\,x}}{e^5}-\frac {2\,a^2\,e^4+4\,a\,c\,d^2\,e^2+2\,c^2\,d^4}{e^5\,\sqrt {d+e\,x}}+\frac {\left (12\,c^2\,d^2+4\,a\,c\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}-\frac {8\,c^2\,d\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2/(d + e*x)^(3/2),x)

[Out]

(2*c^2*(d + e*x)^(7/2))/(7*e^5) - ((8*c^2*d^3 + 8*a*c*d*e^2)*(d + e*x)^(1/2))/e^5 - (2*a^2*e^4 + 2*c^2*d^4 + 4
*a*c*d^2*e^2)/(e^5*(d + e*x)^(1/2)) + ((12*c^2*d^2 + 4*a*c*e^2)*(d + e*x)^(3/2))/(3*e^5) - (8*c^2*d*(d + e*x)^
(5/2))/(5*e^5)

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sympy [A]  time = 17.78, size = 126, normalized size = 1.02 \begin {gather*} - \frac {8 c^{2} d \left (d + e x\right )^{\frac {5}{2}}}{5 e^{5}} + \frac {2 c^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (4 a c e^{2} + 12 c^{2} d^{2}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (- 8 a c d e^{2} - 8 c^{2} d^{3}\right )}{e^{5}} - \frac {2 \left (a e^{2} + c d^{2}\right )^{2}}{e^{5} \sqrt {d + e x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**2/(e*x+d)**(3/2),x)

[Out]

-8*c**2*d*(d + e*x)**(5/2)/(5*e**5) + 2*c**2*(d + e*x)**(7/2)/(7*e**5) + (d + e*x)**(3/2)*(4*a*c*e**2 + 12*c**
2*d**2)/(3*e**5) + sqrt(d + e*x)*(-8*a*c*d*e**2 - 8*c**2*d**3)/e**5 - 2*(a*e**2 + c*d**2)**2/(e**5*sqrt(d + e*
x))

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